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\begin{document}

\title{高等代数二}
\subtitle{8-1-向量的内积 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年3月9日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{8.1.i. 作业：星期天晚上十点半之前在网络教学平台提交 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{enumerate}
\item   整理课堂笔记，补充没写完的计算或证明。
\item   习题(8.1)\#1,2,3,5,6,7, 抄写题目。
\end{enumerate}

\end{frame}

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\begin{frame}{8.1.ii. 目录 }

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\begin{enumerate}
\item[8.1.1.] 欧氏空间的定义
\item[8.1.6.] 欧氏空间的基本性质
\item[8.1.8.] 向量的长度的定义
\item[8.1.9.] 定理8.1.1. 柯西-施瓦茨不等式
\item[8.1.12.] 向量的夹角的定义
\item[8.1.13.] 两个向量是正交的定义
\item[8.1.15.] 定理8.1.2. 向量与子空间之间的正交
\item[8.1.16.] 三角形不等式
\item[8.1.17.] 两个向量的距离的定义
\item[8.1.18.] 欧氏空间的子空间


\end{enumerate}

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\begin{frame}{8.1.iii. 课堂讲解重点 }

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\begin{enumerate}
\item  欧氏空间的概念
\item  向量的长度和夹角的计算
\item  柯西-施瓦茨不等式
\end{enumerate}

\end{frame}

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\begin{frame}{8.1.1. 欧氏空间的定义}

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\begin{itemize}

\item {\color{red}问题：一个欧氏空间是一个实向量空间 $V$ 以及一个内积运算
$$\langle \cdot,\cdot \rangle : V\times V \to \mathbb{R},$$ 
满足哪些公理？}

\item  解答：欧氏空间的内积要满足下述四条公理：

\begin{enumerate}
\item 交换律：对任意的 $\alpha,\beta\in V$, 都有 $\langle \alpha,\beta \rangle = \langle \beta,\alpha \rangle$.
\item 分配律：对任意的 $\alpha,\beta, \gamma \in V$, 都有 $\langle \alpha+\beta,\gamma \rangle = \langle \alpha,\gamma \rangle + \langle \beta,\gamma \rangle$.
\item 数乘：对任意 $\alpha,\beta\in V$, 对任意实数 $k$, 都有 $\langle k\alpha,\beta \rangle = k\langle \alpha,\beta \rangle$.
\item 正定性：对任意非零向量 $\alpha\in V$, 都有 $\langle \alpha,\alpha \rangle>0$. 
\end{enumerate}

\end{itemize}

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\begin{frame}{8.1.2. 例子 }

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\begin{itemize}
\item  {\color{red}问题：在 $n$ 维行向量空间 $V=\mathbb{R}^n$ 中，定义内积运算为 
$$\langle (x_1,x_2,\cdots,x_n), (y_1,y_2,\cdots,y_n) \rangle = x_1y_1+x_2y_2+\cdots+x_ny_n.$$ 
验证这是一个欧氏空间。
}

\item 证明：验证四条公理成立。
%\begin{enumerate}
%\item 
%\item 
%\item 
%\item 
%\end{enumerate}

\end{itemize}

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\begin{itemize}
\item  {\color{red}问题：在 $n$ 维行向量空间 $V=\mathbb{R}^n$ 中，定义内积运算为 
$$\langle (x_1,x_2,\cdots,x_n), (y_1,y_2,\cdots,y_n) \rangle = x_1y_1+2x_2y_2+\cdots+nx_ny_n.$$ 
验证这也是一个欧氏空间。
}

\item 证明：验证四条公理成立。
%\begin{enumerate}
%\item 
%\item 
%\item 
%\item 
%\end{enumerate}

\end{itemize}

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\begin{itemize}
\item  {\color{red}问题：设 $V=C[a,b]$ 是区间 $[a,b]$ 上的一切实数取值的连续函数全体组成的向量空间。
定义内积如下，验证 $V$ 是一个欧氏空间。
$$\langle f,g \rangle = \int_a^b f(x)g(x)dx.$$
}

\item 证明：验证四条公理成立。
%\begin{enumerate}
%\item 
%\item 
%\item 
%\item 
%\end{enumerate}

\end{itemize}

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\begin{frame}{8.1.5. 例子 }

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\begin{itemize}
\item  {\color{red}问题：设 $V$ 是一切平方和收敛的实数序列全体组成的向量空间。对任意两个向量 
$\alpha=(x_1,x_2,\cdots,x_n,\cdots),\, \beta=(y_1,y_2,\cdots,y_n,\cdots),$
定义内积为
$$\langle \alpha,\beta \rangle = \sum\limits_{i=1}^{\infty} x_iy_i. $$
验证这是一个欧氏空间。
}

\item 证明：首先验证这个级数是收敛的，然后验证四条公理成立。
%\begin{enumerate}
%\item 
%\item 
%\item 
%\item 
%\end{enumerate}

\end{itemize}

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\begin{frame}{8.1.6. 零向量与任意向量的内积都是零 }

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\begin{itemize} 

\item  {\color{red}问题：设 $V$ 是一个欧氏空间。设 $\alpha\in V$, 设 $\theta$ 是 $V$ 中的零向量。证明：
$$\langle \theta,\alpha\rangle =0. $$
}

\item 证明：根据 $\theta+\theta=\theta$ 与内积的分配律，可得
$$\langle \theta,\alpha\rangle = \langle \theta+\theta,\alpha\rangle = \langle \theta,\alpha\rangle + \langle \theta,\alpha\rangle.$$
等式两边约去一个 $\langle \theta,\alpha\rangle$ 即得。

%\item 

\end{itemize}

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\begin{frame}{8.1.7. 内积运算在每个位置都保持线性运算 }

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\begin{itemize}

\item  {\color{red}问题：证明在欧氏空间中，成立下述等式
$$\langle \alpha,k_1\beta_1+k_2\beta_2 \rangle 
= k_1\langle \alpha,\beta_1 \rangle + k_2\langle \alpha,\beta_2 \rangle.$$
}

\item 解答：根据公理1（交换律）、公理2（分配律）与公理3（数乘）可证。

\begin{align*}
\langle \alpha,k_1\beta_1+k_2\beta_2 \rangle 
&= \langle k_1\beta_1+k_2\beta_2,\alpha \rangle \\ 
&= \langle k_1\beta_1, \alpha \rangle + \langle k_2\beta_2,\alpha \rangle \\ 
&= k_1\langle \beta_1, \alpha \rangle + k_2\langle \beta_2,\alpha \rangle \\
&= k_1\langle \alpha,\beta_1 \rangle + k_2\langle \alpha,\beta_2 \rangle. 
\end{align*}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{8.1.8. 向量的长度的定义}

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%每页详细内容

\begin{itemize}
\item  {\color{red}问题：什么是欧氏空间中的向量的长度？什么是单位向量？
} 

\item  解答：设 $V$是一个欧氏空间。设 $\alpha\in V$. 定义这个向量的长度为 
$$\lvert \alpha \rvert = \sqrt{\langle \alpha,\alpha \rangle}. $$
若 $\lvert\alpha\rvert =1$, 则称 $\alpha$ 为一个单位向量。

\vspace{0.5cm}

\item {\color{red}问题：设 $\alpha\in V$, 设 $k\in \mathbb{R}$, 则有 $\lvert k\alpha \rvert = \lvert k \rvert \lvert\alpha\rvert$.
}

\item  解答：两边平方，根据内积的运算法则来验证。
$$\lvert k\alpha \rvert ^2 = \langle k\alpha,k\alpha \rangle = k^2 \langle \alpha, \alpha \rangle = k^2 \lvert \alpha \rvert ^2. $$


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{8.1.9. 定理8.1.1. （柯西-施瓦茨不等式）}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $V$ 是一个欧氏空间。设 $\alpha,\beta\in V$, 那么一定有
$$\langle \alpha,\beta \rangle^2 \le \langle \alpha,\alpha \rangle \cdot \langle \beta,\beta \rangle,$$
而且，当且仅当向量组 $\{\alpha,\beta\}$ 线性相关时，等号成立。
}

\item 证明：
\begin{enumerate}
\item  考虑函数 $f(t)=\langle t\alpha+\beta, t\alpha+\beta\rangle$. 
\item  由内积的正定性，可知对任意实数 $t$ 都有 $f(t)\ge 0$. 
\item  根据内积的线性，可得 
$f(t)=\langle \alpha, \alpha\rangle t^2 + 2\langle \alpha, \beta\rangle t + \langle \beta,\beta \rangle$. 
\item  因为二次函数 $f(t)$ 恒非负，所以它的判别式小于等于零，即
\vspace{-0.2cm}
\begin{eqnarray*}
4\langle \alpha,\beta \rangle^2 - 4 \langle \alpha,\alpha \rangle \cdot \langle \beta,\beta \rangle \le 0. 
\end{eqnarray*}

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{8.1.10. 例子 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item {\color{red}问题：将柯西-施瓦茨不等式应用在欧氏空间 $\mathbb{R}^n$, 写出相应的不等式。}
\begin{enumerate}
\item {\color{red} 设 $n=2$, 考虑内积 $\langle (x_1,x_2),(y_1,y_2) \rangle = x_1y_1 + x_2y_2$. }
\item {\color{red} 设 $n=3$, 考虑内积 $\langle (x_1,x_2,x_3),(y_1,y_2,y_3) \rangle = x_1y_1 + 3x_2y_2+5x_3y_3$. }
\end{enumerate}

\item 解答：
\begin{enumerate}
\item  $$(x_1y_1 + x_2y_2)^2 \le (x_1^2 + x_2^2)(y_1^2 + y_2^2). $$ 
\item  $$(x_1y_1 + 3x_2y_2+5x_3y_3)^2 \le (x_1^2 + 3x_2^2+5x_3^2)(y_1^2 + 3y_2^2+5y_3^2). $$
\end{enumerate}


\end{itemize}

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%每页详细内容

\begin{itemize}
\item  {\color{red}问题：将柯西-施瓦茨不等式应用在连续函数全体组成的欧氏空间 $C[a,b]$, 写出相应的不等式。} 

\item 解答：
$$\left( \int_a^b f(x)g(x)dx \right)^2 \le \left( \int_a^b f(x)^2dx \right) \left( \int_a^b g(x)^2dx \right). $$

%\item 

\end{itemize}

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\begin{frame}{8.1.12. 向量的夹角的定义}

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%每页详细内容

\begin{itemize}

\item  {\color{red}问题：欧氏空间中的两个向量的夹角是怎么计算的？}

\item 解答：设 $V$ 是一个欧氏空间。设 $\alpha,\beta \in V$. 
这两个向量之间的夹角 $\theta$ 为 $[0,\pi]$ 之间的一个角度，而且满足下述等式
$$\cos(\theta)=\frac{\langle \alpha,\beta\rangle}{\lvert \alpha \rvert \lvert \beta \rvert}.$$

\item  {\color{red}问题：证明上式右边的取值总是在 $[-1,1]$ 之内。%，从而任意两个向量的夹角是唯一确定的。
}

\item 解答：由柯西-施瓦茨不等式可得。

\end{itemize}

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%每页详细内容

\begin{itemize}

\item  {\color{red}问题：欧氏空间中的两个向量什么时候是正交的？}

\item 解答：若两个向量的内积为零，即 $\langle\alpha,\beta\rangle=0$, 则称这两个向量是正交的。


\end{itemize}

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%每页详细内容

\begin{itemize}

\item  {\color{red}问题：考虑立体空间 $V=\mathbb{R}^3$. 考虑内积为
$$\langle (x_1,x_2,x_3), (y_1,y_2,y_3) \rangle = x_1y_1+x_2y_2+x_3y_3.$$ 
设 $\alpha=(0,1,0), \beta=(1,1,1)$.} 
\begin{enumerate}
\item  {\color{red}用这个内积计算这两个向量的长度与夹角。}
\item  {\color{red}用立体几何的方法计算这两个向量的长度和夹角。}
\end{enumerate}

\item 解答：
\begin{align*}
\cos(\theta) &= \frac{\langle \alpha,\beta\rangle}{\lvert \alpha \rvert \lvert \beta \rvert} 
= \frac{1}{1\cdot \sqrt{3}} = \frac{\sqrt{3}}{3}, \\ 
\theta &= \arccos \frac{\sqrt{3}}{3}. 
\end{align*}

\end{itemize}

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\begin{frame}{8.1.15. 定理8.1.2.}

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%每页详细内容

\begin{itemize}
\item  {\color{red}定理：设 $V$ 是一个欧氏空间。设向量 $\alpha$ 与向量组 $\{\beta_1,\cdots,\beta_m\}$ 中的每一个向量都正交。则向量 $\alpha$ 与这些向量张成的子空间 $L(\beta_1,\cdots,\beta_m)$ 中的每一个向量都正交。
}

\item 证明：
\begin{enumerate}
\item  设 $\xi\in L(\beta_1,\cdots,\beta_m)$, 则存在 $k_1,\cdots,k_m\in\mathbb{R}$, 使得  
$$\xi = k_1\beta_1 + \cdots + k_m\beta_m. $$
\item  根据内积的运算法则，可得
\begin{align*}
\langle \alpha,\xi\rangle & = \langle \alpha, k_1\beta_1 + \cdots + k_m\beta_m \rangle 
 = k_1\langle \alpha, \beta_1\rangle + \cdots + k_m\langle \alpha, \beta_m \rangle \\
& = k_1\cdot 0 + \cdots + k_m\cdot 0 = 0. 
\end{align*}

\end{enumerate}

\item 注：若直线垂直某平面内两条不平行的直线，则垂直该平面内所有直线。

\end{itemize}

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\begin{frame}{8.1.16. 三角形不等式}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}定理：设 $V$ 是一个欧氏空间。设向量 $\alpha,\beta\in V$. 那么一定有
$$\lvert \alpha+\beta \rvert \le \lvert \alpha \rvert + \lvert \beta \rvert. $$ 
}

\item 证明：根据内积的运算法则，以及柯西-施瓦茨不等式，可得 
\begin{align*}
\lvert \alpha+\beta \rvert ^2 
&= \langle \alpha+\beta, \alpha+\beta \rangle \\ 
&= \langle \alpha, \alpha \rangle + 2 \langle \alpha, \beta \rangle + \langle \beta, \beta \rangle \\ 
& \le \lvert \alpha \rvert ^2 + 2\lvert \alpha \rvert \cdot \lvert \beta \rvert + \lvert \beta \rvert ^2 \\ 
&=  (\lvert \alpha \rvert + \lvert \beta \rvert )^2. 
 \end{align*}

\vfill 

\item 注：三角形两边之和大于第三边。

\end{itemize}

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\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：欧氏空间 $V$ 中的两个向量 $\alpha,\beta$ 之间的距离，是指 
$$d(\alpha,\beta) = \lvert \alpha-\beta \rvert,$$
即向量 $\alpha-\beta$ 的长度。写出距离函数 $d(\cdot,\cdot):V\times V\to\mathbb{R}$ 的一些性质。
}

\item 解答：
\begin{enumerate}
\item  两个不同向量之间的距离总是大于零：若 $\alpha\neq \beta$, 则 $d(\alpha,\beta)>0$. 
\item  对称性：$d(\alpha,\beta)=d(\beta,\alpha)$. 
\item  三角形不等式： $d(\alpha,\beta) \le  d(\alpha,\gamma) + d(\gamma,\beta)$. 
\end{enumerate}

\end{itemize}

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\begin{frame}{8.1.18. 欧氏空间的子空间}

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%每页详细内容

\begin{itemize}
\item {\color{red}问题：什么是欧氏空间的子空间？}

\item 解答：称 $W$ 是欧氏空间 $V$ 的子空间，是指：
\begin{enumerate}
\item  作为实向量空间，$W$ 是 $V$ 的向量子空间。
\item  子空间 $W$ 继承了欧氏空间 $V$ 的内积。
\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{8.1.19. 欧氏空间之间的同构}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item {\color{red}问题：写出两个欧氏空间之间的同构的定义。}

\item 解答：
\begin{enumerate}
\item  是两个实向量空间之间的同构。
\begin{enumerate}
\item  是一个线性映射。
\item  是一个双射。
\end{enumerate}
\item  保持内积运算。
\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{习题(8.1)\#1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：设 $V$ 是一个欧氏空间，设 $\xi,\eta\in V$. 
\begin{enumerate}
\item  证明：$\lvert \xi+\eta \rvert ^2 + \lvert \xi-\eta \rvert ^2 = 2\lvert \xi \rvert ^2 + 2\lvert \eta \rvert ^2$. 
\item  证明：$4 \langle \xi, \eta \rangle = \lvert \xi+\eta \rvert ^2 - \lvert \xi-\eta \rvert ^2$. 
\end{enumerate}

}

\item  思路：按照 $\lvert \xi \rvert ^2$ 的定义。考虑定义内积运算的几条公理。


\end{itemize}

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\item  {\color{red}问题：在欧氏空间 $\mathbb{R}^n$ 中， 求向量 $\alpha=(1,1,\cdots,1)$ 与标准基的每个向量
$\varepsilon_i = (0,\cdots,0,1,0,\cdots,0)$ 的夹角。

}

\item  思路：先计算 $n=2,3,4$ 的情形。


\end{itemize}

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\begin{itemize}

\item  {\color{red}问题：在欧氏空间 $\mathbb{R}^4$ 中找出两个单位向量，使它们同时与向量
\begin{eqnarray*}
\alpha = (2,1,-4,0), \,\,
\beta = (-1,-1,2,2), \,\,
\gamma=(3,2,5,4). 
\end{eqnarray*}
中的每一个都正交。 

}

\item  思路：按照两个向量正交的定义，把问题化为求解线性方程组。


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\begin{itemize}

\item  {\color{red}问题：设 $\xi$ 与 $\eta$ 是一个欧氏空间里彼此正交的向量。证明勾股定理：
$$\lvert \xi+\eta \rvert ^2 = \lvert \xi \rvert ^2 + \lvert \eta \rvert ^2. $$

}

\item  思路：按照欧氏空间中的向量的长度的定义，把问题化为内积的运算。


\end{itemize}

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\begin{itemize}

\item  {\color{red}问题：设 $\alpha_1, \alpha_2,\cdots,\alpha_n,\beta$ 都是一个欧氏空间中的向量。
设 $\beta$ 是 $\alpha_1, \alpha_2,\cdots,\alpha_n$ 的线性组合。
设 $\beta$ 与每个 $\alpha_i$ 都正交，$1\le i\le n$. \\  
证明：$\beta=\theta$ 是零向量。
}

\item  思路：按照两个向量正交的定义，把问题化为对内积的运算。


\end{itemize}

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\begin{itemize}

\item  {\color{red}问题：欧氏空间中的向量 $\alpha_1, \alpha_2,\alpha_3$ 的格拉姆行列式定义为
{\footnotesize 
$$
G(\alpha_1, \alpha_2,\alpha_3) = \begin{vmatrix}
\langle \alpha_1, \alpha_1 \rangle & \langle \alpha_1, \alpha_2 \rangle & \langle \alpha_1, \alpha_3 \rangle  \\ 
\langle \alpha_2, \alpha_1 \rangle & \langle \alpha_2, \alpha_2 \rangle & \langle \alpha_2, \alpha_3 \rangle  \\ 
\langle \alpha_3, \alpha_1 \rangle & \langle \alpha_3, \alpha_2 \rangle & \langle \alpha_3, \alpha_3 \rangle  \\ 
\end{vmatrix}. 
$$
}
证明 $G(\alpha_1, \alpha_2,\alpha_3) = 0$ 当且仅当向量组 $\{\alpha_1, \alpha_2,\alpha_3\}$ 线性相关。

}

\item  思路：
\begin{enumerate}
\item  充分性：设 $\alpha_3=k_1\alpha_1+k_2\alpha_2$. 代入右边，计算行列式。
\item  必要性：设格拉姆行列式的第三行可以写成前两行的线性组合。
\end{enumerate}

\end{itemize}

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